1,381,357 research outputs found
Tensor decomposition and homotopy continuation
A computationally challenging classical elimination theory problem is to
compute polynomials which vanish on the set of tensors of a given rank. By
moving away from computing polynomials via elimination theory to computing
pseudowitness sets via numerical elimination theory, we develop computational
methods for computing ranks and border ranks of tensors along with
decompositions. More generally, we present our approach using joins of any
collection of irreducible and nondegenerate projective varieties
defined over . After computing
ranks over , we also explore computing real ranks. Various examples
are included to demonstrate this numerical algebraic geometric approach.Comment: We have added two examples: A Coppersmith-Winograd tensor, Matrix
multiplication with zeros. (26 pages, 1 figure
Computing torus-equivariant K-theory of singular varieties
This expository note surveys some results on equivariant K-theory of
varieties with a torus action, focusing on recent work with Sam Payne and
Richard Gonzales. It is based on my contribution to the Clifford Lectures at
Tulane University in March 2015.Comment: 16 page
The Phillips Machine, The Analogue Computing Traditoin in Economics and Computability
In this paper I try to argue for the desirability of analog computation in economics from a variety of perspectives, using the example of the Phillips Machine. Ultimately, a case is made for the underpinning of both analog and digital computing theory in constructive mathematics. Some conceptual confusion in the meaning of analog computing and its non-reliance on the theory of numerical analysis is also discussed. Digital computing has its mathematical foundations in (classical) recursion theory and constructive mathematics. The implicit, working, assumption of those who practice the noble art of analog computing may well be that the mathematical foundations of their subject is as sound as the foundations of the real analysis. That, in turn, implies a reliance on the soundness of set theory plus the axiom of choice. This is, surely, seriously disturbing from a computation point of view. Therefore, in this paper, I seek to locate a foundation for analog computing in exhibiting some tentative dualities with results that are analogous to those that are standard in computability theory. The main question, from the point of view of economics, is whether the Phillips Machine, as an analog computer, has universal computing properties. The conjectured answer is in the negative.Phillips Machine, Analogue Computation, Digital Computation, Computability, General Purpose Analogue Computer
- …